Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Theoretical and Mathematical Physics

Construction of eigenfunctions for a system of quantum minors of the monodromy matrix for an SL(n,ℂ)-invariant spin chain

P. A. Valinevich, Sergey E. Derkachov, P. P. Kulish, Evgenii Mikhailovich Uvarov

Published Nov 1, 2016• 8 citations
Publisher

Abstract

We consider the problem of seeking the eigenvectors for a commuting family of quantum minors of the monodromy matrix for an SL(n,ℂ)-invariant inhomogeneous spin chain. The algebra generators and elements of the L-operator at each site of the chain are implemented as linear differential operators in the space of functions of n(n−1)/2 variables. In the general case, the representation of the sl n (ℂ) algebra at each site is infinite-dimensional and belongs to the principal unitary series. We solve this problem using a recursive procedure with respect to the rank n of the algebra. We obtain explicit expressions for the eigenvalues and eigenvectors of the commuting family. We consider the particular cases n = 2 and n = 3 and also the limit case of the one-site chain in detail.

DOI
10.1134/s0040577916110015

Related papers

Factorization of \mathcal{R} -matrix and Baxter \mathcal{Q} -operators for genericsl(N) spin chains

Sergey É Derkachov, Alexander N Manashov

We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.

arXiv:0809.2050•Journal of Physics A Mathematical and Theoretical•Jan 21, 2009•25 citations

R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain

Sergey E. Derkachov

The problem of constructing the SL(N, C) invariant solutions to the Yang-Baxter equation is considered.The solutions (R-operators) for arbitrarily principal series representations of SL(N, C) are obtained in an explicit form.We construct the commutative family of the operators Q k (u) which can be identified with the Baxter operators for the noncompact SL(N, C) spin magnet.

arXiv:nlin/0612003•Symmetry Integrability and Geometry Methods and Applications•Dec 2, 2006•37 citations

Iterative construction of eigenfunctions of the monodromy matrix for $SL(2,\mathbb {C})$ magnet

S É Derkachov, Alexander N Manashov

Eigenfunctions of the matrix elements of the monodromy matrix provide a convenient basis for studies of spin chain models. We present an iterative method for constructing the eigenfunctions in the case of the SL(2,C) spin chains. We derived an explicit integral representation for the eigenfunctions and calculated the corresponding scalar products (Sklyanin's measure).

arXiv:1401.7477•Journal of Physics A Mathematical and Theoretical•Jul 10, 2014•29 citations

Construction of Monodromy Matrix in the F- basis and scalar products in Spin chains.

А. А. Овчинников

We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin- 1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the Monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F- basis and the usual basis. 1. Introduction. One of the most important open problems in the theory of quantum integrable models is the calculation of the correlation functions. In the framework of the Algebraic Bethe Ansatz method [1] the problem is the combinatorial complexity of calculations due to the structure of Bethe eigenstates. Only the very limited number of physical results for the correlation

•Jan 1, 2000•3 citations

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

Kun Hao, Junpeng Cao, Guang-Liang Li +3 more

An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.

arXiv:1601.04771•Journal of High Energy Physics•May 1, 2016•4 citations

Quantum spin chains and classical integrable systems

Zabrodin Anton

Abstract This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.

•Aug 10, 2017•0 citations
We consider the problem of seeking the eigenvectors for a commuting family of quantum minors of the monodromy matrix for an SL(n,ℂ)-invariant inhomogeneous spin chain. The algebra generators and elements of the L-operator at each site of the chain are implemented as linear differential operators in the space of functions of n(n−1)/2 variables. In the general case, the representation of the sl n (ℂ) algebra at each site is infinite-dimensional and belongs to the principal unitary series. We solve this problem using a recursive procedure with respect to the rank n of the algebra. We obtain explicit expressions for the eigenvalues and eigenvectors of the commuting family. We consider the particular cases n = 2 and n = 3 and also the limit case of the one-site chain in detail.
We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.
Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C)SL(2,\mathbb {C})SL(2,C) magnet
We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin- 1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the Monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F- basis and the usual basis. 1. Introduction. One of the most important open problems in the theory of quantum integrable models is the calculation of the correlation functions. In the framework of the Algebraic Bethe Ansatz method [1] the problem is the combinatorial complexity of calculations due to the structure of Bethe eigenstates. Only the very limited number of physical results for the correlation
An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.
Abstract This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.